Ever since Feynman et al showed that the transitions between two quantum states were equivalent to the spin excitations in NMR experiments, there has been great interest in developing the optical analogues of NMR, particularly because of the power of NMR to probe complex systems with great selectivity. The invention of the laser stimulated a great deal of work to develop the optical analogues because the laser provided the required coherent source.

Early experiments closely followed NMR methods. π/2 and π pulses were delivered to samples by carefully controlling the phase and intensity of the excitation. It was difficult to preserve the phase information of the coherences because the vibrational and electronic coherences dephase ~9 orders of magnitude more quickly than NMR coherences because of the stronger interactions with the environmental thermal bath. These dephasing rates are much larger than typical Rabi frequencies, so it was difficult to create coherences with π/2 and π pulses owing to the time required being longer than that permitted by the dephasing times. Success was achieved only by making the pulses stronger so the Rabi periods were shorter or increasing the dephasing times so the dephasing times were comparable to the Rabi period. Typically, the dephasing times were increased by cooling the sample to extremely low temperatures. Although these approaches allowed limited studies, they did not provide a viable approach for measuring typical samples.

Coherent multidimensional spectroscopy (CMDS) has emerged as a technique with widespread use because it was discovered that phase matching and time ordering of the excitation pulses could isolate particular coherence pathways just as well as supplying a series of phased π/2 and π pulses. For example, a molecule in its ground state can interact with a field to create a coherence either through absorption on its ket- or bra-side. However, if the experiment uses a stimulated photon echo phase matching geometry $\vec{k}_4 = -\vec{k}_1 + \vec{k}_2 + \vec{k}_3$, the first interaction must be a bra-side absorption; so for a two state system, the pathways are restricted to the two shown in figure 5.

CMDS can be based on time domain or frequency domain methods. In time domain methods, one directly measures the temporal oscillations of the coherences and then performs a Fourier transform to obtain a frequency spectrum. If excitation pulses were available that were very short compared with the period of a coherence’s oscillation, the measurement could be performed by scanning the delay of the excitation pulse relative to the coherence being measured and one would immediately obtain the temporal phase oscillations of the coherence. Unfortunately, the excitation pulses are not that short so one must use a local oscillator. A local oscillator usually is a fourth field that is almost identical to the signal beam. The electric fields of the local oscillator and the signal heterodyne so the intensity depends on $\left| E_{LO} + E_{S} \right| ^2$. If the time delay between the local oscillator and the signal is changed, one directly measures the oscillations of their relative phases. A Fourier transform will provide the frequency domain spectrum. If the delay between the first two excitation beams changes, the signal beam will reflect the changes in phase of the coherence created by the first beam and these in turn will be measured by the local oscillator as well. Fourier transform of these variations provide a second frequency axis for the 2D-IR spectroscopy. If the first two excitation beams create a coherence, the delay between the second and third excitation beams will scan the temporal phase changes of that intermediate coherence and Fourier transformation will provide a third frequency dimension. It is very important that the phase relationships between the excitation beams remains stable over the entire measurement time. This factor requires that the multiple excitation beams be derived from a single source that acts to define the phase of each beam. It also requires that the relative path lengths within the experimental system are interferometrically stable to a small fraction of a wavelength. Finally, in a pure time domain  experiment, all the quantum states are excited impulsively and one measures all of the coherences simultaneously through the temporal oscillations. Practically, the range of quantum states that can be measured depends on the excitation pulse bandwidth. Typically, the pulse widths are ~50-100 fs and these bandwidths excite quantum states over a range of ~150-300 cm^-1. 

In frequency domain methods, one scans the frequency of the excitation beams while monitoring the intensity of the output coherence. The intensity is enhanced by each resonance and the enhancements are multiplicative. Phase coherence is again required but it is only required during the pulse sequence. Long term phase coherence is not required. Finally, in a pure frequency domain experiment, only specific quantum states are excited at any one time so individual coherence pathways and specific states are selected. The range is limited only by the tunability of the excitation sources. There are advantages to working in a mixed frequency/time domain where the excitation pulses are long enough to excite individual quantum states and avoid the difficulties involved with maintaining long term phase coherence but they are also short enough that one can measure the temporal dynamics of the coherences and populations of the quantum states.